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Omni-MATH / Let ABC be an acute triangle. Let DAC,EAB, and FBC be isosceles triangles exterior to ABC, with DA=DC,EA=EB, and FB=FC, such that ∠ADC=2∠BAC, ∠BEA=2∠ABC, ∠CFB=2∠ACB. Let D' be the …

Problem

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problem

Let ABCABC be an acute triangle. Let DAC,EABDAC,EAB, and FBCFBC be isosceles triangles exterior to ABCABC, with DA=DC,EA=EBDA=DC, EA=EB, and FB=FCFB=FC, such that ∠ADC=2∠BAC,∠BEA=2∠ABC,∠CFB=2∠ACB. \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. Let $D'$ be the intersection of lines DBDB and EFEF, let $E'$ be the intersection of ECEC and DFDF, and let $F'$ be the intersection of FAFA and DEDE. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]
Plain-text mathematical notation (without MathML)
Let ABC be an acute triangle.  Let DAC,EAB, and FBC be isosceles triangles exterior to ABC, with DA=DC,EA=EB, and FB=FC, such that
∠ADC=2∠BAC, ∠BEA=2∠ABC, ∠CFB=2∠ACB.
Let $D'$  be the intersection of lines DB and EF, let $E'$ be the intersection of EC and DF, and let $F'$   be the intersection of FA and DE. Find, with proof, the value of the sum
\[
\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.
\]
Original LaTeX notation
Let $ABC$ be an acute triangle.  Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that
\[
\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad
\angle CFB = 2 \angle ACB.
\]
Let $D'$  be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$   be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum
\[
\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.
\]

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Source and history

task_id
091ad14e-2c4b-5c58-9324-3aa24c169fe4
task_key
test--091ad14e-2c4b-5c58-9324-3aa24c169fe4
task_revision_id
1
config / split
/ test

Official source

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